Activities
Students re-represent a state given in Dirac notation in matrix notation
Problem
You are given the following Gibbs free energy: \begin{equation*} G=-k T N \ln \left(\frac{a T^{5 / 2}}{p}\right) \end{equation*} where \(a\) is a constant (whose dimensions make the argument of the logarithm dimensionless).
Compute the entropy.
Work out the heat capacity at constant pressure \(C_p\).
Find the connection among \(V\), \(p\), \(N\), and \(T\), which is called the equation of state (Hint: find the volume as a partial derivative of the Gibbs free energy).
- Compute the internal energy \(U\).
Find an expression for the free energy as a function of \(T\) of a system with two states, one at energy 0 and one at energy \(\varepsilon\).
From the free energy, find expressions for the internal energy \(U\) and entropy \(S\) of the system.
Plot the entropy versus \(T\). Explain its asymptotic behavior as the temperature becomes high.
Plot the \(S(T)\) versus \(U(T)\). Explain the maximum value of the energy \(U\).
Treat the ground state of a quantum particle-in-a-box as a periodic function.
Set up the integrals for the Fourier series for this state.
Which terms will have the largest coefficients? Explain briefly.
Are there any coefficients that you know will be zero? Explain briefly.
Using the technology of your choice or by hand, calculate the four largest coefficients. With screen shots or otherwise, show your work.
- Using the technology of your choice, plot the ground state and your approximation on the same axes.
Problem
At low temperatures, a diatomic molecule can be well described as a rigid rotor. The Hamiltonian of such a system is simply proportional to the square of the angular momentum \begin{align} H &= \frac{1}{2I}L^2 \end{align} and the energy eigenvalues are \begin{align} E_{\ell m} &= \hbar^2 \frac{\ell(\ell+1)}{2I} \end{align}
What is the energy of the ground state and the first and second excited states of the \(H_2\) molecule? i.e. the lowest three distinct energy eigenvalues.
At room temperature, what is the relative probability of finding a hydrogen molecule in the \(\ell=0\) state versus finding it in any one of the \(\ell=1\) states?
i.e. what is \(P_{\ell=0,m=0}/\left(P_{\ell=1,m=-1} + P_{\ell=1,m=0} + P_{\ell=1,m=1}\right)\)At what temperature is the value of this ratio 1?
- At room temperature, what is the probability of finding a hydrogen molecule in any one of the \(\ell=2\) states versus that of finding it in the ground state?
i.e. what is \(P_{\ell=0,m=0}/\left(P_{\ell=2,m=-2} + P_{\ell=2,m=-1} + \cdots + P_{\ell=2,m=2}\right)\)
Students, working in pairs, use their left arms to represent each component in a two-state quantum spin 1/2 system. Reinforces the idea that quantum states are complex valued vectors. Students make connections between Dirac, matrix, and Arms representation.
The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.
Give the general solution of the differential equation: \[\frac{d^2 u}{d\phi^2}+u=0\]
It is NOT necessary to show any work.
Give the general solution of the differential equation: \[\frac{d^2 \Phi}{d\phi^2}+7\Phi=0\]
It is NOT necessary to show any work.
Give the general solution of the differential equation: \[\frac{d^2 y}{dx^2}+Ay=0\] Make sure that you can give the solution of this equation regardless of the geometric names of the dependent and independent variables and for either sign for the constant \(A\).
It is NOT necessary to show any work. You may NOT, however, give a solution that has a negative number inside a square root. I am testing whether you can recognize this equation and remember its solution. This equation comes up over and over again in physics, but disguised by different symbols. I am also testing whether you recognize that the geometric character of the equation changes depending on the sign of \(A\).
Problem
Consider a phase transformation between either solid or liquid and gas. Assume that the volume of the gas is way bigger than that of the liquid or solid, such that \(\Delta V \approx V_g\). Furthermore, assume that the ideal gas law applies to the gas phase. Note: this problem is solved in the textbook, in the section on the Clausius-Clapeyron equation.
Solve for \(\frac{dp}{dT}\) in terms of the pressure of the vapor and the latent heat \(L\) and the temperature.
Assume further that the latent heat is roughly independent of temperature. Integrate to find the vapor pressure itself as a function of temperature (and of course, the latent heat).
In this activity students use the known speed of earthquake waves to estimate the Young's modulus of the Earth's crust.
Gauss's Law: \[ \oint \vec{E}\cdot \hat{n}\, dA = {1\over\epsilon_0}\, Q_{\hbox{enc}} \]
Ampère's Law:
\[ \oint \vec{B}\cdot d\vec{r} = \mu_0 \, I_{\hbox{enc}} \]
Potentials: \begin{eqnarray*} \vec{E}&=&-\vec{\nabla} V\\ \vec{B}&=&\vec{\nabla}\times\vec{A} \end{eqnarray*}
Maxwell's Equations: \begin{eqnarray*} \vec{\nabla}\cdot\vec{E} &=& \frac{\rho}{\epsilon_0}\\ \vec{\nabla}\cdot\vec{B} &=& 0\\ \vec{\nabla}\times\vec{E} &=& 0\\ \vec{\nabla}\times\vec{B} &=& {\mu_0}\, \vec{J} \end{eqnarray*}
Superposition Laws: \begin{eqnarray*} V(\vec{r}) &=& \frac{1}{4\pi\epsilon_0} \int{\rho(\vec{r}')\, d\tau'\over \vert \vec{r}-\vec{r}'\vert}\\ \vec{E}(\vec{r}) &=& \frac{1}{4\pi\epsilon_0} \int{\rho(\vec{r}')(\vec{r}-\vec{r}')\, d\tau'\over \vert \vec{r}-\vec{r}'\vert^3}\\ \vec{A}(\vec{r}) &=& \frac{\mu_0}{4\pi} \int{\vec{J}(\vec{r}')\, d\tau'\over \vert \vec{r}-\vec{r}'\vert}\\ \vec{B}(\vec{r}) &=& \frac{\mu_0}{4\pi} \int{\vec{J}(\vec{r}')\times (\vec{r}-\vec{r}')\, d\tau'\over \vert \vec{r}-\vec{r}'\vert^3}\\ V(B)-V(A)&=&-\int_A^B \vec{E}\cdot d\vec{r} \end{eqnarray*}
Position Vectors \begin{align*} \vec{r} &= x \hat{x} + y\hat{y} + z\hat{z}\\ &= s \hat{s} + z\hat{z}\\ &= r\hat{r} \end{align*}The distance between two position vectors
- In cylindrical coordinates: \[\left\vert\vec r -\vec r^{\prime}\right\vert =\sqrt{s^2+s^{\prime\, 2}-2s\, s^{\prime}\cos(\phi- \phi^{\prime}) +(z-z^{\prime})^2}\]
- In spherical coordinates: \[\left\vert\vec r -\vec r^{\prime}\right\vert =\sqrt{r^2+r^{\prime\, 2}-2r\, r^{\prime}\left[ \sin\theta\sin\theta^{\prime}\cos(\phi-\phi^{\prime}) +\cos\theta\cos\theta^{\prime}\right]}\]
Rectangular Coordinates: \begin{eqnarray*} \vec{\nabla} f &=& \frac{\partial f}{\partial x}\,\hat{x} + \frac{\partial f}{\partial y}\,\hat{y} + \frac{\partial f}{\partial z}\,\hat{z} \\ \vec{\nabla}\cdot\vec{F} &=& \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \\ \vec{\nabla}\times\vec{F} &=& \left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z}\right)\hat{x} + \left(\frac{\partial F_x}{\partial z} -\frac{\partial F_z}{\partial x}\right)\hat{y} + \left(\frac{\partial F_y}{\partial x} -\frac{\partial F_x}{\partial y}\right)\hat{z} \end{eqnarray*}
Cylindrical Coordinates: \begin{eqnarray*} \vec{\nabla} f &=& \frac{\partial f}{\partial s}\,\hat{s} + \frac{1}{s}\frac{\partial f}{\partial \phi}\,\hat{\phi} + \frac{\partial f}{\partial z}\,\hat{z} \\ \vec{\nabla}\cdot\vec{F} &=& \frac{1}{s}\frac{\partial}{\partial s}\left({s}F_{s}\right) + \frac{1}{s}\frac{\partial F_\phi}{\partial \phi} + \frac{\partial F_z}{\partial z} \\ \vec{\nabla}\times\vec{F} &=& \left( \frac{1}{s}\frac{\partial F_z}{\partial \phi} - \frac{\partial F_\phi}{\partial z} \right) \hat{s} + \left(\frac{\partial F_s}{\partial z}-\frac{\partial F_z}{\partial s}\right) \hat{\phi} + \frac{1}{s} \left( \frac{\partial}{\partial s}\left({s}F_{\phi}\right) - \frac{\partial F_s}{\partial \phi} \right) \hat{z} \end{eqnarray*}
Spherical Coordinates: \begin{eqnarray*} \vec{\nabla} f &=& \frac{\partial f}{\partial r}\,\hat{r} + \frac{1}{r}\frac{\partial f}{\partial \theta}\,\hat{\theta} + \frac{1}{r\sin\theta}\frac{\partial f}{\partial \phi}\,\hat{\phi} \\ \vec{\nabla}\cdot\vec{F} &=& \frac{1}{r^2}\frac{\partial}{\partial r}\left({r^2}F_{r}\right) + \frac{1}{r\sin\theta}\frac{\partial}{\partial \theta}\left({\sin\theta}F_{\theta}\right) + \frac{1}{r\sin\theta}\frac{\partial F_\phi}{\partial \phi} \\ \vec{\nabla}\times\vec{F} &=& \frac{1}{r\sin\theta} \left( \frac{\partial}{\partial \theta} \left({\sin\theta}F_{\phi}\right) - \frac{\partial F_\theta}{\partial \phi} \right) \hat{r} + \frac{1}{r} \left( \frac{1}{\sin\theta} \frac{\partial F_r}{\partial \phi} - \frac{\partial}{\partial r}\left({r}F_{\phi}\right) \right) \hat{\theta} \\ && \quad + \frac{1}{r} \left( \frac{\partial}{\partial r}\left({r}F_{\theta}\right) - \frac{\partial F_r}{\partial \theta} \right) \hat{\phi} \end{eqnarray*}
Lorentz Force Law:\[\vec{F}=q_{\hbox{test}}\left(\vec{E}+\vec{v}\times\vec{B}\right)\]
Step and Delta Functions: \begin{eqnarray*} \frac{d}{dx} \theta(x-a)&=&\delta(x-a)\\ \int_{-\infty}^{\infty} f(x)\delta(x-a)\, dx&=&f(a) \end{eqnarray*}
Vector Calculus Theorems: \begin{eqnarray*} \oint \vec{F} \cdot d\vec{A} &=& \int \vec{\nabla} \cdot \vec{F} d\tau\\ \oint \vec{F} \cdot d\vec{\ell} &=& \int (\vec{\nabla} \times \vec{F}) \cdot d\vec{A}\\ \end{eqnarray*}
Total Charge and Current: \begin{eqnarray*} Q &=& \int \rho (\vec{r}') d\tau'\\ I &=& \int \vec{J} (\vec{r}') \cdot d\vec{A'}\\ \end{eqnarray*}
Students use Tinker Toys to represent each component in a two-state quantum spin system in all three standard bases (\(x\), \(y\), and \(z\)). Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT change the state of the system) and relative phase (which does change the state of the system). This activity is optional in the Arms Sequence Arms Sequence for Complex Numbers and Quantum States.
This activity acts as a reintroduction to doing quantum calculations while also introducing the matrix representation on the ring, allowing students to discover how to index and form a column vector representing the given quantum state. In addition, this activity introduces degenerate measurements on the quantum ring and examines the state after measuring both degenerate and non-degenerate eigenvalues for the state.
Students consider the dimensions of spin-state kets and position-basis kets.
In this activity, students apply the Stefan-Boltzmann equation and the principle of energy balance in steady state to find the steady state temperature of a black object in near-Earth orbit.
Students consider projectile motion of an object that experiences drag force that in linear with the velocity. Students consider the horizontal motion and the vertical motion separately. Students solve Newton's 2nd law as a differential equation.
In this small group activity, students work out the steady state temperature of an object absorbing and emitting blackbody radiation.
Students work in a small group to write down an equation for a travelling wave.
The instructor and students do a skit where students represent quantum states that are “measured” by the instructor resulting in a state collapse.
These are notes, essentially the equation sheet, from the final review session for https://paradigms.oregonstate.edu/courses/ph441.
This handout lists Motivating Questions, Key Activities/Problems, Unit Learning Outcomes, and an Equation Sheet for a Unit on Classical Mechanics Orbits. It can be used both to introduce the unit and, even better, for review.
Students work out heat and work for rectangular paths on \(pV\) and \(TS\) plots. This gives with computing heat and work, applying the First Law, and recognizing that internal energy is a state function, which cannot change after a cyclic process.
Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.
Students use an applet to explore the role of the parameters \(N\), \(x_o\), and \(\sigma\) in the shape of a Gaussian \begin{equation} f(x)=Ne^{-\frac{(x-x_0)^2}{2\sigma^2}} \end{equation}
Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.